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Unformatted text preview: Chapter 1
Introduction
Spacecraft dynamics and control is a rich subject involving a variety of topics from
mechanics and control theory. In a ﬁrst course in dynamics, students learn that the
motion of a rigid body can be divided into two types of motion: translational and
rotational. For example, the motion of a thrown ball can be studied as the combined
motion of the mass center of the ball in a parabolic trajectory and the spinning motion
of the ball rotating about its mass center. Thus a ﬁrst approximation at describing the
motion of a ball might be to model the ball as a point mass and ignore the rotational
motion. While this assumption may give a reasonable approximation of the motion,
in actuality the rotational motion and translational motion are coupled and must be
studied together to obtain an accurate picture of the motion. The motion of the mass
center of a curve ball is an excellent example of how the rotational and translational
motions are coupled.1 The ball does not follow the parabolic trajectory predicted by
analysis of the translational motion, because of the unbalanced forces and moments
on the spinning ball.
The study of spacecraft dynamics is similar to the study of baseball dynamics.
One ﬁrst gains an understanding of the translational motion of the mass center using
particle dynamics techniques, then the rotational motion is studied. Thus the usual
study of spacecraft dynamics begins with a course in orbital dynamics, usually in the
junior or senior year, and is followed by a course in attitude dynamics in the senior
year or the ﬁrst year of graduate study. In this book I assume that the student has had
a semester of orbital dynamics, but in fact I only use circular orbits, so that a student
with some appreciation for dynamics should be able to follow the development without
the orbital dynamics background. The basics of orbital dynamics are included in
Appendix A, and the text by Bate, Mueller, and White2 is an aﬀordable introduction
to the required material.
Another important way to decompose dynamics problems is into kinematics and
kinetics. For translational motion, kinematics is the study of the change in position
for a given velocity, whereas kinetics is the study of how forces cause changes in
velocity. For rotational motion, kinematics is the study of the change in orientation
11 Copyright Chris Hall January 12, 2003 12 CHAPTER 1. INTRODUCTION for a given angular velocity, and kinetics is the study of how moments cause changes in
the angular velocity. Translational kinematics is relatively easy to learn, since it only
involves the motion of a point in threedimensional space. Rotational kinematics,
however, is usually more diﬃcult to master, since it involves the orientation of a
reference frame in threedimensional space.
Kinetics
Force
aﬀects
Moment aﬀects Velocity
Angular Velocity aﬀects
Position
aﬀects Orientation Kinematics
In this chapter, I begin by describing how attitude dynamics and control arises
in the operation of spacecraft. This discussion is followed by a description of the
fundamental attitude control concepts that are in widespread use. Finally, I give an
overview of the textbook. 1.1 Attitude dynamics and control in operations Essentially all spacecraft include one or more subsystems intended to interact with or
observe other objects. Typically there is one primary subsystem that is known as the
payload. For example, the primary mirror on the Hubble Space Telescope is one of
many instruments that are used to observe astronomical objects. The communications
system on an Intelsat satellite is its payload, and the infrared sensor on board a
Defense Support System (DSP) satellite is its payload. In each case, the payload must
be pointed at its intended subject with some accuracy speciﬁed by the “customer”
who purchased the spacecraft. This accuracy is typically speciﬁed as an angular
quantity; e.g., 1 degree, 10 arcseconds, or 1 milliradian. The attitude control system
designer must design the attitude determination and control subsystem (ADCS) so
that it can meet the speciﬁed accuracy requirements.
It costs more than $10,000 to put a kilogram of mass into lowEarth orbit (LEO),
and even more to put it into geostationary orbit (GEO).3 A typical spacecraft masses
about 500 kg, and costs tens to several hundred millions of dollars to design, manufacture, test, and prepare for launch. All of this money is spent to purchase the
mission capability of the spacecraft. The attitude control system, propulsion system,
launch vehicle, and so forth, are only there so that the mission may be performed
eﬀectively. If the mission can be accomplished without an ADCS, then that mass can
be used to increase the size of the payload, decrease the cost of launch, or in some
other way improve the performance or reduce the cost. The bottom line is that the
payload and its operation are the raison d’ˆtre for the spacecraft. This focus justiﬁes
e Copyright Chris Hall January 12, 2003 1.1. ATTITUDE DYNAMICS AND CONTROL IN OPERATIONS 13 our spending a little time on describing how the ADCS ﬁts into the operations of the
payload.
There are many spacecraft payloads, but most ﬁt into one of just two categories:
communications and remote sensing. On a communications satellite, the payload
comprises the radio transceivers, multiplexers, and antennas that provide the communications capability. Historically, most communications satellites have been in the
geostationary belt, and have been either dualspin or threeaxis stabilized. More recently, a host of LEO commsats has been put into orbit, including the 66satellite
Iridium constellation, the 36satellite OrbComm constellation, and the planned 48satellite GlobalStar constellation. The Iridia use hydrazine propulsion for threeaxis
stabilization, whereas the OrbComms are gravitygradient stabilized. The GlobalStar
spacecraft are threeaxis stabilized, using momentum wheels, magnetic torquers, and
thrusters. In commsats, the mission of the ADCS is to keep the spacecraft pointed
accurately at the appropriate ground station. The more accurate the ADCS, the more
tightly focused the radio beam can be, and the smaller the power requirements will
be. However, ADCS accuracy carries a large price tag itself, so that design trades are
necessary.
There are two basic types of remote sensing satellites: Earthobserving and spaceobserving. An Earthobserving spacecraft could be nadirpointing, it could be scanning the land or sea in its instantaneous access area, or it could selectively point to
and track speciﬁc ground targets. In the ﬁrst case, a passive gravitygradient stability
approach might suﬃce, whereas in the second and third cases, an active ADCS would
likely be required, using some combination of momentum wheels, magnetic torquer
rods, or thrusters.
A spaceobserving system could simply point away from the Earth, with an additional requirement to avoid pointing at the sun. This “antiEarth” pointing is essentially the ADCS requirement for the CATSAT mission being built by the University
of New Hampshire∗ . The CATSAT ADCS uses momentum wheels and magnetic
torquer rods. More complicated spaceobserving systems require both largeangle
slewing capability and highly accurate pointing control. The Hubble Space Telescope
is a wellknown example. It uses momentum wheels for attitude control, and performs largeangle maenuvers at about the same rate as the minutehand of a clock.
The HST does not use thrusters because the plume would contaminate the sensitive
instruments.
One problem with momentum wheels, reaction wheels, and control moment gyros
is momentum buildup: external torques such as the gravity gradient torque and solar
radiation pressure will eventually cause the wheel to reach its maximum speed, or
the CMG to reach its maximum gimbal angle. Before this saturation occurs, the
spacecraft must perform an operation called momentum unload, or momentum dump:
external torques are applied, using thrusters of magnetic torquer rods, that cause the
∗
Cooperative Astrophysics and Technology SATellite.
http://www.catsat.sr.unh.edu/. Copyright Chris Hall January 12, 2003 See Refs. 4 and 5, and the website 14 CHAPTER 1. INTRODUCTION ADCS to decrease the wheels’ speeds or the CMGs’ gimbal angles. Depending on the
spacecraft, this type of “maneuver” may be performed as often as once per orbit.
Another ADCS operation involves keeping the spacecraft’s solar panels pointing
at the sun. For example, when the HST is pointing at a particular target, it still has
a degree of freedom allowing it to rotate about the telescope axis. This rotation can
be used to orient the solar panel axis so that it is perpendicular to the direction to the
sun. Then the panels are rotated about the panel axis so that they are perpendicular
to the sun direction. This maneuver is known as yaw steering. 1.2 Overview of attitude dynamics concepts The attitude of a spacecraft, i.e., its orientation in space, is an important concept
in spacecraft dynamics and control. Attitude motion is approximately decoupled
from orbital motion, so that the two subjects are typically treated separately. More
precisely, the orbital motion does have a signiﬁcant eﬀect on the attitude motion, but
the attitude motion has a less signiﬁcant eﬀect on the orbital motion. For this reason
orbital dynamics is normally covered ﬁrst, and is a prerequisite topic for attitude
dynamics. In a third course in spacecraft dynamics, the coupling between attitude
and orbital motion may be examined more closely. In this course, we focus on the
attitude motion of spacecraft in circular orbits, with a brief discussion of the attitude
motion of simple spacecraft in elliptic orbits.
Operationally, the most important aspects of attitude dynamics are attitude determination, and attitude control. The reason for formulating and studying the dynamics problem is so that these operational tasks can be performed accurately and
eﬃciently. Attitude determination, like orbit determination, involves processing observations (“obs”) to obtain parameters for describing the motion. As developed in
Appendix A, we can determine the orbit of the satellite if we have the range (ρ),
˙
range rate (ρ), azimuth (Az), azimuth rate (Az), elevation (El), and elevation rate
˙
˙
(El) from a known site on the Earth. The deterministic algorithm to compute the six
orbital elements from these six measurements is wellknown and can be found in most
astrodynamics texts. Of course, due to measurement noise, the use of only six measurements is not practical, and statistical methods are normally used, incorporating
a large number of observations.
Similarly, we can determine the attitude, which can be described by three parameters such as Euler angles, by measuring the directions from the spacecraft to some
known points of interest. For example, suppose a spacecraft has a Sun sensor and an
Earth sensor. The two sensors provide vector measurements of the direction from the
spacecraft to the sun (vs ) and to the Earth (ve ). These are normally unit vectors, so
each measurement provides two pieces of information. Thus the two measurements
provide four known quantities, and since it only takes three variables to describe attitude, the problem is overdetermined, and statistical methods are required (such as
least squares). Actually there is a deterministic method that discards some of the Copyright Chris Hall January 12, 2003 15 1.2. OVERVIEW OF ATTITUDE DYNAMICS CONCEPTS
Table 1.1: Attitude Control Concepts
Concept
Gravity Gradient
Spin Stabilization
DualSpin
Momentum Wheels
Control Moment
Gyros
Magnetic
Torquer Rods
Thrusters
Dampers Passive/
Active
P
A
A
A
A Internal/ Environmental
External
E
Y
I
N
I
N
I
N
I
N A E Y A
P or A E
I N
N measurements and we develop it in Chapter 4. A wide variety of attitude determination hardware is in use. The handbook edited by Wertz (Ref. 6) provides a wealth of
information on the subject. The more recent text by Sidi (Ref. 7) is notable for its
appendices on hardware speciﬁcations.
Controlling the attitude of a spacecraft is also accomplished using a wide variety
of hardware and techniques. The choice of which to use depends on the requirements
for pointing accuracy, pointing stability, and maneuverability, as well as on other
mission requirements such as cost and lifetime. All attitude control concepts involve
the application of torques or moments to the spacecraft. The various methods can
be grouped according to whether these torques are passive or active, internal or
external, and whether the torques are environmental or not. A reasonably complete
list of concepts is shown in Table 1.1.
The most fundamental idea in the study of attitude motion is the reference frame.
Throughout the book we work with several diﬀerent reference frames, and you must
become familiar and comfortable with the basic concept. As a preview, let us consider
an example where four diﬀerent reference frames are used. In Fig. 1.1, we show three
reference frames useful in describing the motion of a spacecraft in an equatorial orbit
about the Earth. One of the reference frames whose origin is at the center of the
ˆ
Earth, O, is an inertial reference frame with unit vectors ˆ, J, and K. This frame is
Iˆ
ˆ axis points in the
usually referred to as the Earthcentered inertial (ECI) frame. The I
direction of the vernal equinox, and the ˆ J plane deﬁnes the equatorial plane. Thus
Iˆ
ˆ axis is the Earth’s rotation axis, and the Earth spins about K with angular
ˆ
the K
velocity ω⊕ = 2π radians per sidereal day. In vector form, the angular velocity of the
ˆ
Earth is ω⊕ = ω⊕ K.
The other frame centered in the Earth is an Earthcentered, Earthﬁxed (ECEF) Copyright Chris Hall January 12, 2003 16 CHAPTER 1. INTRODUCTION ˆˆ
K, K ˆ
o1 ˆ
o3
O
ˆ
o2
ˆ
J ˆ
I ˆ
J
ˆ
I Figure 1.1: EarthCentered Inertial (ECI = ˆ, J, K ), EarthCentered EarthFixed
Iˆ ˆ
ˆˆˆ
Iˆˆ
(ECEF = ˆ , J , K ), and Orbital Reference Frames (Fo = {o1 , o2 , o3 }) for an Equatorial Orbit Copyright Chris Hall January 12, 2003 17 1.2. OVERVIEW OF ATTITUDE DYNAMICS CONCEPTS ˆˆ
K, K ˆ
o1
ˆ
o3
ˆ
o2 O ˆ
J
ˆ
I
ˆ
J
ˆ
I
Figure 1.2: EarthCentered Inertial, EarthCentered EarthFixed, and Orbital Reference Frames for an Inclined Orbit Copyright Chris Hall January 12, 2003 18 CHAPTER 1. INTRODUCTION ˆ
frame which rotates with respect to the ECI frame with angular velocity ω ⊕ = ω⊕ K.
ˆ , J , and K . Note that K is the same in
ˆ
ˆ
ˆ
This frame has unit vectors represented by I
both the ECI and ECEF frames. The importance of the ECEF frame is that points on
the surface of the Earth, such as ground stations and observation targets, are ﬁxed in
this frame. However, since the frame is rotating, it is not an inertial reference frame.
The other frame in the ﬁgure has its origin at the mass center of the spacecraft.
This point is assumed to be in an orbit (circular or elliptical) about the Earth, thus its
motion is given. As drawn in the ﬁgure, this orbit is also an equatorial orbit, so that
ˆ
the orbit normal is in the K direction. The origin of this frame is accelerating and so
it is not inertial. This frame is called the orbital frame because its motion depends
ˆˆ
ˆ
only on the orbit. The unit vectors of the orbital frame are denoted o1 , o2 , and o3 .
ˆ
The direction pointing from the spacecraft to the Earth is denoted by o3 , and the
ˆ
ˆ
direction opposite to the orbit normal is o2 . The remaining direction, o1 is deﬁned
ˆ
ˆ
ˆ
ˆ
by o1 = o2 × o3 . In the case of a circular orbit, o1 is in the direction of the spacecraft
velocity vector. For those familiar with aircraft attitude dynamics, the three axes
of the orbital frame correspond to the roll, pitch, and yaw axes, respectively. This
reference frame is noninertial because its origin is accelerating, and because the frame
is rotating. The angular velocity of the orbital frame with respect to inertial space
ˆ
is ωo = −ωo o2 . The magnitude of the orbital angular velocity is constant only if the
orbit is circular, in which case ωo = µ/r3 , where µ is the gravitational parameter,
and r is the orbit radius (see Appendix A). If the orbit is not circular, then ω o varies
with time. Note well that ωo is the angular velocity of the orbital frame with respect
to the inertial frame and is determined by the translational, or orbital, dynamics.
Another reference frame of interest is shown in Fig. 1.3 in relation to the orbital
ˆˆ
ˆ
frame. This frame is the bodyﬁxed frame, with basis vectors b1 , b2 , and b3 . Its
origin is at the spacecraft mass center, just as with the orbital frame. However, the
spacecraft body, or platform, is in general not aligned with the orbital frame. The
relative orientation between these two reference frames is central to attitude determination, dynamics, and control. The relative orientation between the body frame
and the orbital frame is determined by the satellite’s rotational dynamics, which is
governed by the kinetic and kinematic equations of motion. The primary purpose of
this text is to develop the theory and tools necessary to solve problems involving the
motion of the body frame when the orbit is known. 1.3 Overview of the textbook Most textbooks on this subject begin with some treatment of kinematics and then
proceed to a study of a variety of dynamics problems, with some control problems
perhaps included. Our approach is similar, but our aim is to spend more time up
front, both in motivation of the topics, and in developing an understanding of how to
describe and visualize attitude motion. To this beginning, we have an introductory Copyright Chris Hall January 12, 2003 1.3. OVERVIEW OF THE TEXTBOOK Figure 1.3: Orbital and Body Reference Frames Copyright Chris Hall January 12, 2003 19 110 CHAPTER 1. INTRODUCTION chapter on Space Mission Analysis that will hopefully help readers to develop an
appreciation for how attitude dynamics ﬁts into the overall space mission. For a
more traditional course, this chapter could be read quickly or even skipped entirely.
Chapter 3 introduces attitude kinematics, developing the classical topics in some
detail, and introducing some new topics that may be used in a second reading. Chapter 4 covers the important topic of attitude determination. This topic is not usually
covered in an introductory course, but I believe that mastery of this subject enhances
the student’s appreciation for the remaining material. In Chapter 5 we develop the
standard equations of motion and relevant results for rigid body dynamics. These
topics lead directly to Chapter 6 on satellite attitude dynamics, where we apply basic
dynamics principles to a variety of problems. In Chapter 7 we introduce and develop equations of motion for the gyroscopic instruments that are used as sensors in
spacecraft attitude control systems. These are used in some simple examples, before proceeding to Chapter 8 where more rigorous development of attitude control
problems is presented. 1.4 References and further reading Dynamics and control of artiﬁcial spacecraft has been the subject of numerous texts
and monographs since the beginning of the space age. Thomson’s book,8 originally
published in 1961, is one of the earliest, and is currently available as a Dover reprint.
The book remains a valuable reference, despite its age. Wertz’s handbook6 is perhaps
the best reference available on the practical aspects of attitude determination and
control. The text by Kaplan9 treats a wide range of topics in both orbital and attitude dynamics. Kane, Likins, and Levinson10 present a novel approach to satellite
dynamics, using Kane’s equations. Hughes’ book11 focuses on modeling and analysis of attitude dynamics problems, and is probably the best systematic and rigorous
treatment of these problems. Rimrott’s book12 is similar to Hughes,11 but uses scalar
notation, perhaps making it more accessible to beginning students of the subject.
Wiesel13 covers rigid body dynamics, as well as orbital dynamics and basic rocket dynamics. Agrawal’s book14 is designoriented, but includes both orbital and attitude
dynamics. The brief book by Chobotov15 covers many of the basics of attitude dynamics and control. Especially useful are the “Recommended Practices” given at the end
of each chapter. The books by Griﬃn and French,16 Fortescue and Stark,17 Larson
and Wertz,18 and Pisacane and Moore19 are all designoriented, and so present useful
information on the actual implementation of attitude determination and control systems, and the interaction of this subsystem with the overall spacecraft. The chapter
on attitude in Pisacane and Moore gives an indepth treatment of the fundamentals,
whereas the relevant material in Larson and Wertz is more handbookoriented, providing useful rules of thumb and simple formulas for sizing attitude determination and
control systems. Bryson20 treats a variety of spacecraft orbital and attitude control
problems using the linearquadratic regulator technique. Sidi7 is a practiceoriented Copyright Chris Hall January 12, 2003 BIBLIOGRAPHY 111 text, providing many detailed numerical examples, as well as current information on
relevant hardware. The new book by Wie21 furnishes a modern treatment of attitude
dynamics and control topics. Finally, an excellent source of information on speciﬁc
spacecraft is the Mission and Spacecraft Library, located on the World Wide Web at
http://leonardo.jpl.nasa.gov/msl/home.html. Bibliography
[1] Neville de Mestre. The Mathematics of Projectiles in Sport, volume 6 of Australian Mathematical Society Lecture Series. Cambridge University Press, Cambridge, 1990.
[2] Roger R. Bate, Donald D. Mueller, and Jerry E. White. Fundamentals of Astrodynamics. Dover, New York, 1971.
[3] John R. London III. LEO on the Cheap — Methods for Achieving Drastic Reductions in Space Launch Costs. Air University Press, Maxwell Air Force Base,
Alabama, 1994.
[4] Chris Whitford and David Forrest. The CATSAT attitude control system. In
Proceedings of the 12th Annual Conference on Small Satellites, number SSC98IX4, Logan, Utah, September 1998.
[5] Ken Levenson and Kermit Reister. A high capability, low cost university satellite
for astrophysical research. In 8th Annual Conference on Small Satellites, Logan,
Utah, 1994.
[6] J. R. Wertz, editor. Spacecraft Attitude Determination and Control. D. Reidel,
Dordrecht, Holland, 1978.
[7] Marcel J. Sidi. Spacecraft Dynamics and Control: A Practical Engineering Approach. Cambridge University Press, Cambridge, 1997.
[8] W. T. Thomson. Introduction to Space Dynamics. Dover, New York, 1986.
[9] Marshall H. Kaplan. Modern Spacecraft Dynamics & Control. John Wiley &
Sons, New York, 1976.
[10] Thomas R. Kane, Peter W. Likins, and David A. Levinson. Spacecraft Dynamics.
McGrawHill, New York, 1983.
[11] Peter C. Hughes. Spacecraft Attitude Dynamics. John Wiley & Sons, New York,
1986.
[12] F. P. J. Rimrott. Introductory Attitude Dynamics. SpringerVerlag, New York,
1989. Copyright Chris Hall January 12, 2003 112 BIBLIOGRAPHY [13] William E. Wiesel. Spaceﬂight Dynamics. McGrawHill, New York, second edition, 1997.
[14] Brij N. Agrawal. Design of Geosynchronous Spacecraft. PrenticeHall, Englewood
Cliﬀs, NJ, 1986.
[15] V. A. Chobotov. Spacecraft Attitude Dynamics and Control. Krieger Publishing
Co., Malabar, FL, 1991.
[16] Michael D. Griﬃn and James R. French. Space Vehicle Design. AIAA Education
Series. American Institute of Aeronautics and Astronautics, Washington, D.C.,
1991.
[17] Peter W. Fortescue and John P. W. Stark, editors. Spacecraft Systems Engineering. John Wiley & Sons, Chichester, 1991.
[18] Wiley J. Larson and James R. Wertz, editors. Space Mission Analysis and Design. Microcosm, Inc., Torrance, CA, second edition, 1995.
[19] Vincent L. Pisacane and Robert C. Moore, editors. Fundamentals of Space Systems. Oxford University Press, Oxford, 1994.
[20] Arthur E. Bryson, Jr. Control of Spacecraft and Aircraft. Princeton University
Press, Princeton, 1994.
[21] Bong Wie. Space Vehicle Dynamics and Control. AIAA, Reston, Virginia, 1998. 1.5 Exercises 1. What types of attitude control concepts are used by the following spacecraft?
If you can, tell what types of sensors and actuators are used in each case.
(a) Explorer I
(b) Global Positioning System
(c) Hubble Space Telescope
(d) Intelsat IV
(e) Iridium
(f) OrbComm
(g) Starshine
(h) Cassini
(i) HokieSat Copyright Chris Hall January 12, 2003 1.5. EXERCISES 113 2. What companies manufacture the following attitude control actuators? List
some of the performance characteristics for at least one speciﬁc component
from each type of actuator.
(a) momentum wheels
(b) control moment gyros
(c) magnetic torquer rods
(d) damping mechanisms
(e) hydrazine thrusters
3. What companies manufacture the following attitude determination sensors?
List some of the performance characteristics for at least one speciﬁc component from each type of sensor.
(a) Earth horizon sensors
(b) magnetometers
(c) rate gyros
(d) star trackers
(e) sun sensors
4. Which control strategies rely on naturally occurring ﬁelds, and what ﬁelds are
they?
5. For a circular orbit, what are the directions of the position, velocity, and orbital
angular momentum vectors in terms of the orbital frame’s base vectors? Support
your answer with a reasonably accurate sketch.
6. Repeat Exercise 5 for an elliptical orbit.
7. Make a sketch of the reference frames missing from Fig. 1.3. Copyright Chris Hall January 12, 2003 ...
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